NASA Technical Reports Server (NTRS) 19810015206: On the structures and mapping of auroral electrostatic potentials

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Space Sciences Laboratory
Report No. SSL-81 (7951 )-l

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ON THE STRUCTURES AND MAPPING OF
AURORAL ELECTROSTATIC POTENTIALS

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by

Y. T. Chiu and A» L. Newnan

Space Sciences Laboratory
The Aerospace Corporation
El Segundo, California 90245

J. M. Cornwall

Department of Physics
University of California at Los Angeles
Los Angeles, California

I

This work was supported in part by NASA Solar-Terrestrial Theory
Program Contract NASW-3434 and in part by NASA Headquarters

Contract NASW-3327.

I

Abstract

We examine the mapping of magnetospherlc and ionospheric electric fields
in a kinetic model of magnetospheric-ionospheric electrodynamic coupling
proposed for the aurora by Chiu and Cornwall (1980). A new feature is the
generalization of the kinetic current-potential relationship to the return-
current region (identified as a region where the parallel potential drop from
magnetosphere to ionosphere is positive); such a return current always exists
unless the ionosphere is electrically charged to grossly unphysical values.
We are able for the first time to give a coherent phenomenological picture of
both the low-energy return current and the high-eneigy precipitation of an
inverted-V. The mapping between magnetospherlc and ionospheric electric
fields is phrased in terms of a Green's function which acts as a filter,
emphasizing magnetospherlc latitudinal spatial scales of order (when mapped to
the ionosphere) 50-150 km. This same length, when multiplied by perpendicular
electric fields just above the ionosphere, sets the scale for parallel
potential drops between the ionosphere and equatorial magnetosphere.

2

I

INTRODUCTION

As the result of particle and field observations In the aurora by rocket
and satellite-borne Instrunents, it is by now fairly well-establishe 1 that the
auroral particles are accelerated by steady (relative to particle transit
time) electric potential differences of ~ I-IO kilovolts between the magneto-
sphere and ionosphere, aligned along the magnetic field [Evans, 1974; Croley
et al., 1978; Mlzera and Fennell, 1977; Shelley et al«, 1976; Mozer et al.,
1977]. These observations confirmed and refined earlier indications of pos-
sible particle acceleration in the aurora [Frank and Ackerson, 1971; Gurnett
and Frank, 1973]. Consequently, recent theoretical efforts have been focused
on the formation of the auroral electric acceleration potential by various
kinetic mechanisms [e.g.. Swift, 1975; Kan, 1975; Hudson and Mozer, 1978;
Lemaire and Scherer, 1974; Chiu and Schulz, 1978; Stern, 1981] which are to be
contrasted with auroral models based on MHD considerations [e.g., Sato, 1978;
Miura and Sato, 1980; Goertz and Boswell, 1979]. Each of these two categories
of auroral models ten^s to Ignore what is most important in the other category
and is correspondingly incomplete [Chiu et al., 1980, 1981]. In particular,
most of the kinetic auroral models omit ionospheric and/or cross-field charge-
separation effects, which amounts to decoupling neighboring magnetic field
lines thus yielding no definite connection between parallel and perpendicular
electric fields.

Recently Chiu and Cornwall [1980] Initiated a program to remedy these
defects, generalizing the kinetic models to account for ionospheric current
conservation and charge conservation in the magnetosphere. These authors
wrote down a simple differential equation in L relating the ionospheric poten-
tial to the L-dependence of the parallel potential drop between Ionosphere and
magnetosphere, which, in effect, specifies the mapping of magnetospheric and

3

ionospheric electric fields along laagnetlc field lines in the presence of
parallel electric potential drops. This equation followed from the nearly
linear relation between the current of precipitated auroral electrons at the
ionosphere and the magnetosphere-ionosphere potential drop [Chiu and Cornwall,
1980; Fridman and Lemaire, 1980] predicted by kinetic theory when the mirror
ratio is large. Lyons [1980, 1981] has also studied this differential equa-
tion in some detail, motivated in part by observations [Lyons et al., 1979]
which confirm the linear current-potential relationship, in connection with
the convection reversal boundary. (Chiu et al. [1980] have also noted the
connection between auroral parallel potential drop and the convection reversal
boundary). More recently, Kan and Lee [1980] have studied the problem of
momentum transfer from ionosphere to magnetosphere with similar ideas.

In this paper we ‘“port on numerical and analytic investigation of elec-
trostatic field mapping in the presence of parallel potential drops between
the magnetosphere and ionosphere. In particular, we are able to consider the
effects of boundary conditions such tnat the ionosphere Is not grossly charged
up. In effect, this is the condition that there is no net Pedersen current in
or out of the auroral zone (assumed to circle the earth). Under these condi-
tions, there is always a return current region. Moreover, the parallel poten-
tial drop in the return current region is typically < (10-25) % of the central
potential drop, so the return current is carried by relatively low-energy
electrons, say tens to a few hundred eV.

We show that the general solution of our model admits potential struc-
tures which can generate both S-shape (perpendicular electric field enhance-
ment without field reversal) and V-shape (with field reversal) "shock” struc-
tures [Temerln et al., 1981], depending primarily upon the boundary conditions
assumed. Indeed, we show that the V-shaped potential structure does not

4

differ materially from the S-shaped structure except that imposed boundary
conditions are such Lhat in the V-sbape the magnetospheric and ionospheric
potential extrema are forced to lie on the same magnetic field line. This
view of the relationship between the two classes of potential structures lends
Itself to the interpretation that one should reasonably tee higher probability
of occurrence for S-shapes than V-shapes. A systematic classification of
model solutions and their implications on auroral return currents are given in
Sections IV and V.

An important consequence of -or Green's function formulation as iono-
spheric potential response to an Imposed magnetospherlc potential is that it
can be directly used to map electrostatic potentials from the equator to the
ionosphere by relating perpendicular electric fields to those associated with
a kinetic-model magnetic field-aligned pontential drop and current. The
mapping of electric fields in the magnetosphere [e.g., Mozer, 197C] and in the
ionosphere-atmosphere [e.g., Chiu, 1974] is a very important problem for
auroral electrodynamic observations and interpretations [e.g., Mozer, 1971).
The magnetospherlc-ionospheric mapping problem for the latitudinal component
of is illustrated in Figure 1. Before the advent of parallel potential
drops, it was assumed that magnetic field lines were electric equipotentlals
(because of assumed infinite parallel conductivity); hence, electric field
mapping between the magnetosphere and ionosphere was strictly geometrical
depending on the distance between neighboring field lines. In other words,
the s.:ales of ionospheric electric fields were related to that of the magneto-
spheric fields strictly by the geometric convergence of the magnetic field, as
required by - 0; i.e., the line Integral of ^ along the magnetosphere-
ionosphere circuit reduces to in Fig. 1. However, if a field-
aligned potential drop such as in Fig. 1 exists, the line integral

5

of 2 over the magnetosphere-ionosphere circuit not only involves

also the perpendicular scale of the field-aligned currents J| since the

mapping depends on where the field lines and L 2 are located in relation to

the upward and downward field-aligned currents. Roughly speaking, the scale

of parallel potential drops in the auroral region is found by multiplying

perpendicular equatorial magnetospheric electric fields by the usual geometric

3/2

mapping factor (■ L ) and by the scale length (50-100 km) of inverted-V
precipitation regions. In a later work we will take up the problem of large-
scale mapping in quantitative detail.

6

11. CHARGE AND CURRENT CONSERVATION IN A KINETIC MODEL

In this section we give a brief sununary of a kinetic model formulation of
magneto8phere~ionosphere coupling leading to auroral acceleration [Chiu and
Cornwall, 19dU]. The basic premise is that auroral particle distributions are
in quasi'static collisionless equilibrium (for time scales long compared to
ion transit time) with the electric and magnetic fields. It has been pointed
out by many authors [see review by Stern, 1981] that differential pitch-angle
anisotropy between electrons and ions in a dipolar flux tube would lead to a
magnetic field-aligned electric potential drop of several kilovolts even in a
one-dimensional model in which the effects of the perpendicular electric field
are ignored. Such one-dimensional models produce features of particle distri-
bution functions in velocity space in agreement with S3-3 particle observa-
tions (Chiu and Schulz, 1978). In addition, such one dimensional models
predict that the magnetic field-aligned current density Jj should be approxi-
mately proportional to the magnetic field-aligned potential difference between
the ionosphere and the magnetosphere [Fridman and Lemaire, 1980; Chiu and
Cornwall, 1980). This relationship is in agreement with rocket observations
[Lyons et al., 1979).

Chiu and Cornwall [1980] generalized such kinetic models to two-
dimensions to include the influences of the perpendicular electric field by
invoking kinetic charge conservation in the auroral region (Poisson's equa-
tion) and current conservation in a schematic sheet-like ionosphere with
height-integrated Pedersen conductivity Thus, for ionospheric potential

(f, the height-integrated current conservation equation states that

V. • (E„ V) - -J, , (O

-L p X

7

where J| Is defined to be negative for downgoing electrons. Now the kinetic
models (or rocket observations) Imply, aside from a term to be Identified with
diffuse auroral precipitation, that Jj Is proportional to the magnetic field-
aligned difference between ^ and the electric potential at the magnetos pherlc
equator

- J, “ Q (♦ - ^ q ) (2)

where Q > 0 depends on particle densities and velocities. Equation (2) has
been v'rltten down by several authors for a bl-Maxwelllan distribution [Chiu
and Cornwall, 1980; Fridman and Lemalre, 1980]. This remarkable linear rela-
tion between and ♦ ~ holds because of the smallness of the mirror
ratio Bq/B^. In the Appendix we give the generalization of (2) to an arbi-

trary distribution function, which shows that the parameter ^ Is of order
2 - -

Ne /Mv, where v Is a velocity typical of the given distribution function.
Within the approximation made In the present paper, the properties of magneto-
spheric particles appear directly only in the parameter Q; auroral features
are otherwise determined by magnetospherlc perpendicular electric fields and
by the Ionospheric Pedersen conductivity (these latter quantities, of course,
may be In part determined by the particle parameters).

Combining (1) and (2) one obtains an equation specifying the ionospheric
electric-potential response ^ to a given magnetospherlc dynamo potential

* ^^p " Q

Lyons (1980) has studied this equation, choosing to represent "discontinu-
ities in the magnetospherlc convection electric field f with < 0." Lyons*

8

solutions show no return current region, and are based on a boundary condition
of constant t at Infinity, both at the Ionosphere and at the magnetospherlc
equator. In this paper we classify and Interpret solutions of (3) using the
Green's function technique, and Imposing the condition that the Ionosphere
does not become electrically charged to grossly unphyslcal values; our solu-
tions are thus different from Lyons'. ^

Of course, equation (3) by Itself tells us nothing about what happens
between the equatorial magnetosphere and the top of the ionosphere. The
physics of this region Is largely governed by Poisson's equation and the
relation between net charge density, electrostatic potential, and magnetic
mirroring forces. In the presence of an Inhomogeneous magnetic field,
Poisson's equation reads [Chiu and Cornwall, 1980]:

V • (Ki?,) + B |- (B~^ E.) - 4ire (N - N ) (4)

1 1 ds I 1 e

where are complicated functions of B and and K » 1 is the plasma

dielectric constant, which depends on N^, N^, B and We will not use (4)
directly In the present work; for us the Important consequence of (4) Is that
field lines are coupled, In the magnetosphere, over lengths scaled by the
Larmor radius. On substantially larger length scales, such as concern us In
this paper, field-line coupling is dominated by equation (3).

Below the top of the Ionosphere (say, ~ 2000 km), the physics of the
ele'-trlc field Involves ionization and recombination processes, as well as
rollislonal conductivities In the E-reglon (where most of the ionospheric
current associated with auroras Is flowing). We have not considered the
physics of this region In any detail, mostly L>cause it Is very complicated.
To achieve the phenomenological approach used here, we need only note that

s

9

Ionospheric return currents are generated in the colllstonal E“region, and
that Poisson's equation allows us to relate Ej In this region to that In the

magnetosphere. In the usual way, we express an Ignorance of the detailed

processes going on between the E~reglon and 2000 km by Integrating over this
range of altitude, j in (1) and (3). In this paper where there Is no need to
distinguish the E-reglon from the rest of the Ionosphere, we adopt the conve-
nient (but Inpreclue) terminology of referring to quantities with sub-

script A as ionospheric. When there Is need for a precise distinction, we use
the subscript 1 to denote the E-reglon ionospere and A for 4 U'»ntitles evalu-
ated at 2000 km (the baropause).

In using equations (3) and (4) we assume all quantities depend only on
the coordinate x, the horizontal distance In the north-south direction at the
baropause (s"A, where s is the distance along the field line from the

equator). Of course, the magnetospherlc potential ♦q is originally given as a
function of latitudinal distance at the equator Xg.; they are related by x ■ Xg
(Bq/B,) , and ♦q - 4>(j[x(Bj/3q) ]. This means that when we speak of

raagnetospheric perpendicular electric fields, these fields are scaled
geometrically to the ionosphere as if there were no parallel potential drop.

Originally equation (3) was derived for the case when the ionospheric
potential ♦(x) was greater than the magnetospheric potential ♦q(x) on a given
field line, for only then could the relation -Jj « ♦“♦q derived. (J| < 0

corresponds to downgoing electrons.) The reason is that the derivation of
this relation depends on the smallness of the inverse mirror ratio (Bq/B^);
this ratio enters because the distribution function of the precipitating
electrons is originally specified at the equator, but evaluated at the iono-
sphere. A similar argument is not directly applicable to the return current,
which has its source in the ionosphere. But other arguments, given below,

10

allow us to conclude that -Jj is still linear in ♦“♦q, even when this poten-
tial drop is reversed in sign, although the (positive) factors of proportiona-
lity are not necei^sarily the same for upgoing and downgoing We thus

generalize the current-potential relation to

-Jj - Q(x) (♦(x) - ♦q(x)! (5)

throughout the whole auroral region and for both signs of current; Q > 0 may
depend on x both implicitly and explicitly, e.g«, Q may assume different
values for ^ - ♦q > 0 and for ^ - ♦q < 0.

It is actually a question of some delicacy whether, in (5), 4 - means

~ ov - i.. In contrast, this is not at issue for the same relation
t 0 I 0

(2) used in the precipitation region, because is much larger than

that ib, ♦o ~ * ^0 ” ^I upward current region of electron

precipitation. This is not so for the return current region, and exactly what
we mean by ♦ - ♦q In l5) affects the value of Q. Since we do not know very
well what Q is in the return current region we leave this question open in our
parametric studies of (3).

The return current, frequently observed to lie adjacent to the upward
current of auroras le.g., Kamide et al., 1979], is formed by conversirn of
directly-precipitating electrons, and their secondaries, into horizontal
Pedersen current at altitudes ^170 km (above 200 km, the Federsen conductivi-
ty drops rapidly). These Pedersen currents carry a net negative charge tc the
edges of the precipitation region, which thus acquires a potential suitable
for expelling ionospheric electrons upward along the magnetic field* (There
are not enough magnet ospheric ions to support the alternate scenario of ion
precipitation [Lui et al., 1977],) The actual charge imbalance is very small.

n

only a tiny fraction of the charge carried In by precipitation. For example,
a downward electron flux F, If not removed fr<xn the Ionosphere promptly by
return currents, produces a surface charge density at the Ionosphere. The
associated electric field produced by F over a time t Is given by E| AxeFt.
If F ■ lO^e/cm^sec, t ■ I sec, we find Ej ■ 200 kV/m! Since in fact E^ is
considerably less than 1 mV/ra at the Ionosphere, the net charge density of the
Ionosphere is less than 5 e/cm^. Although this Is a tiny charge Imbalance, It
Is directly responsible for the return current flow. In our calculations we
need not deal directly with the net charge density; It is clearly adequate to
insist on current conservation, co that all the charge that flows Into the
ionosphere flows out again. Note that this condition Is violated in Lyons'
(1980, 1981) calculations.

Now we must relate the return current density to the electric field
produced by the charge imbalance of the Ionosphere. Since the return current
is created In the collisional E-reglon, the relation between current and field
is the usual one: J| ■ 0 | E|. Actually, this should be integrated over the

various altitudes at which conversion of Pedersen current to field-aligned
current takes place; we do not knov the details of this process, so Instead we
employ height-integrated quantities:

C6)

Here is an effective height-averaged collision frequency, and h Is the
altitude difference between the collisional Ionosphere and the regime where
collisions are ineffective (roughly the i^'opause). Above the baropause, J|
is given by a geometric scaling law expressing the openlng-up of flux tubes:

12

( 7 )

• < A: J,(«) - J,(t) 1^

Furcheraore, for ■ < £, E|(t) lx given by the same scaling as in (7). To see
this, return to Poisson's equation (4), averaged over a horizontal dlj-
tance 4x larger than tlM ion Lansor radius (a few kb), but seall conpared to
the size of the return current region (50-700 km). The first term on the left
of (4) Is small after averaging, and we drop it. The charge density (right-
hand side of (4)} is likewise small, since magnetic mirror forces do not act
to separate the low-energy electrons aikl ions. One then concludes that

s < A: E|(s)

B(s) . ~ V B(s)

in the return current region. Of course, (6) - (8) together tell us that J|

~ (4j - 4^) everywhere along the line, while (8) can be Integrated over s
from 0 to A to give (4g " 4^) as varying linearly with 4j ~ 4j. • Then (5), the
proportionality of current to 4 " 4q, is established for the return current
region. As we have said, the constant of proportionality depends on whether
by 4 we mean 4^ 4|*

In summary, the "mapping" of a given magnetospheric potential distribu-
tion 4q(x) to the ionospheric E-region (where the potential distribution
is 4(x)) in the presence of field-aligned potential drop distribution 4(x) -
4q(x) Is governed by

where F is the dimensionless profile of the height-integrated ionospheric
Pedersen conductivity ^^^(x) = Eq F(x) such that F(dr")"l. The function x(x) =

13

1 n

[Q(x)/£q] > 0 Is an Inverse scale length set by the magnetosphere-

ionosphere coupling. From (9), It Is clear that this natural scale acts as a
filter as the distribution ♦q(x) Is "mapped" Into ♦(x). The solution of (9)
under various physical restraints, such as (4), will be the main topic tu be
dealt with In this paper.

We have already said that the Ionosphere Is slightly charged, but by an
extremely tiny amount (the precipitated charge Is relieved by the return
current). Equally negligible Is the net charge of the magnetosphere-

ionosphere system. Integrated for x ■ - * to •. It thus follows from
Poisson's equation (4), Integrated over all x with neglect of the right-hand
side, that

Ej^(-) - Ej^(— ) - 0 - ♦'(-) (10)

By integrating (4) from -“ to •, we learn that

/ dx x^x) [♦(x) - 4 q(x)] - ♦’(-) - ♦•(— ) - 0 (11)

«C0

This shows Immediately that ^(x) - ^q(x) must change sign; the crossover point
Xp, where ♦(x^) ■ ^q(x^). Is the bounds’^y between the region of direct elec-
tron precipitation and the return current.

14

III. GENERAL FEATURES OF ELECTRIC FIELD MAPPING

As has been alluded to previously, a major purpose of this paper is to
analyze the dependence of the ionospheric electric potential response upon the
Lnposed magnetospheric dynamo potential distribution and upon the boundary and
charge constraint conditions assumed. For purposes of establishing the con-
sistency of boundary values for the general scheme formulated by Chiu and
Cornwall [1980], it is convenient to consider the case of constant (but dis-
continuous) parameters F and ic in (9). On the practical side, since the
entire problem with constant parameters can be solved analytically, the re-
sults of this section provide the basis for analysis without the encumbrances
of a computational effort. We shai 1 show in the next section that our con-
clusions are not basically altered when F and < are made functions of x.

For constant parameters (F ■ 1, k ■ constant) in (9), the general solu-
tion can be written as

❖ (x)

,, -<x +tcx
te + Ue

- f / * dy ♦o(y) f /, dy ♦„(y)."‘'‘->'> (O)

where the determination of constants (C, D) and the Integration limits depend
on boundary <ondltions. Note that (13) is written in terms of the general
one-dimensional Green's function solution for given source function hence,
the lnteri>r.*tatlon that (|i is the ionospheric response to Because < ■

can be different constants in the upward and downward current
regions, the complete solution for ^ with a given set of values of k, must be
made continuous at the boundary of the two regions; indeed, this procedure can
be applied for any number of regions of different x values.

15

As an example, let us consider the explicit solution specified by the
following conditions:

a. The ionospheric potential ♦(x) and the dynamo potential ♦y(x) are
synmetrir about the origin; thus ♦(x) “ ♦(-x), so we need only consider the
domain 0 x < •.

b. The domain 0 ^ x < •• is split into two regions defined by different
constant values of F and k. Because these parameters are assumed constant, we
can set F - 1 and (9) is specified by a single parameter ic^ (i » 1, 2) in each
region. The boundary between the two regions is labeled x ■ x^,. Thus, (9)
becomes

♦

1,2

- V

(14)

with D ^ X ^ Xp labeled as region 1 and x^, ^ x < • labeled as region 2.

c. The total integrated charge of the ionosphere is assumed zero. The
symmetry assumption a above Implies that total charge in 0 x < “ also
vanishes, as expressed in (10). Adopting the notation of that equation, we
have

♦^(-) - ♦[(O) - 0 (15)

Now because of the symmetry assumption, ♦j(O) must be an extremum,
i.e., (^1 (0) ■ 0. Therefore, the effects of assumptions a, b and c correspond
to the boundary conditions

♦J(0) - ♦ 2 ^-) - 0

(16)

16

d. At the Interface x ■ x^., we require the continuity of ^ and its
derivative (a discontinuity in would imply a surface charge layer at x^.):

tj(x^) - 07)

♦l(x^) - ♦•(x^) (18)

Up to this point, we have treated the interfacial point x ■ x^, as if
given; but, by virtue of its definition as the Interface between regions of
upward and downward current, we have by application of (5) at x^

where the second equality of (19) is redundant with (17). In addition to the
four boundary conditions (16) - (18), which determine the set of four coeffi-
cients, (Cj, C2, Dj, D2) in terms of x^, (19) is a transcendental equation for
x^. To render the procedure more explicit, we write the regional solutions to
(14)

-K.x K.x K. X K.(y-x) K. X -►..(yx)

i|i.(x)-C,e + D e + 7“ / dy ♦„(y)e ~ y / dy 4>^(y)e (20)

0 0

-KjX <2 X “ -K5(y-x)

(((^(x) - C^e + — / dy <CQ(y)e +— / dy ♦jj(y)e (21)

*" X X

c

from which (16) - (19) can be applied to determine the unknown constants in

terras of the parameters and ^6e procedure is straightforward for an

assumed ^^(x).

As an example, we show in Fig. 1 a solution ^ for a given of the form

17

♦q(x) - a le “ (a/b) e ***]

( 22 )

where a”^ ■ 7b. 5 km, b”^ ■ 73.5 km and A is a normalization so chosen that
^q(O)* A(l-a/b) < 0 is equivalent to ten divisions of the ordinate of Fig.
2. The scale length of is (because a is nearly equal to b) about 160 km;
whereas the natural scale lengths and K 2 ^ are respectively 73 km and 33
km. From Fig. 2, the distance to the cross-over point is about 160 km,
while the return current extends for some distance past that.

It Is clear from (20) and (21) that the scale length associated with the
ionospheric potential ^(x) depends on the as well as on the scale associ-
ated with This correlation of scales is analogous to electric field
mapping in the collislonal ionosphere (Chiu, 1974), where finite but different
parallel and perpendicular conductivities play the same roles as our field-
aligned and Pedersen currents. As in the case of ionospheric electric field
mapping, ‘t would be convenient to have a "rule of thumb" for the convolution
of scales of the potential drop. For this propose, we consider the case of a
dynamo potential of scale y

♦q(x) - Ae"^* (23)

in the simplest situation in which " •'2 " This has a discontinuity
in at X ■ 0, which makes it somewhat artificial.

An easy calculation yields the cross-over distance x^,:

\ - (y-x)"* tn (y/k) (24a)

18

If, Instead of taking - ic^ “ f 2 " " (which forces ^(x) *

♦Q(x)for X > Xj.), x^. Is Increased by a factor of two over the value given by
(24a). The opposite extreme of ® yields scale lengths in the range 1.4-2
times Xj, in (24a), depending on y/x. That is to say x^ varies by at most a
factor of tvro when <2 varied.

An important scale length is that of the potential difference ^
since it is a measurable quantity. From (13) and (23) we find this scale
length to be

- *^0
(y + <) ♦ ■

(24b)

Note that as y + 0 the cross-over distance x^ approaches infinity, while
X. -*■ X this is the case considered by Lyons (1980). (In later work, Lyons
(1981) has considered finite y.) In our simple example x^ is independent of
potentials, but x^ depends on them. We estimate at x"0 that

- ^q(^~^q) ^ lies between 1 and 3 for typical cases, so (24) shows that x^ is
governed by y ^ in the limit of large y, but tends to x ^ for small y. That
is, srall-scale magnetospheric structure can be transmitted down to the
ionosphere with little change, but if the magnetospheric scale length (of
course, mapped geomatrically onto the ionosphere) is large compared
to X \ the scale length of the Inverted-V region tends to x It is

important that small-scale magnetospheric structures are not filtered out,
since they nuv well be responsible for small-scale (in Lamor radius) effects
observed in the auroral ionosphere [e.g.. Swift, 1979; Lysak and Carlson,
1981).

Now we come to one of the most important features of equation (14) and
its associated boundary conditions: The central potential drop Af = ^(0)-

19

(liy(O) is not an undetermined parameter; instead it is set by the scale
length K and by the magnetospheric Ej^q(x), geometrically mapped onto the
ionosphere. This means that the large-scale convection field mapping problem
can be solved with relatively minor modifications to the solution for zero
parallel potential drop. It is not our purpose to discuss this large-scale
mapping in detail here, so we simplify to the case of constant < to make our
point. It is then an easy matter to integrate (20) by parts and find:

OO

A4» = 4>(0) -• 4>q( 0) ■ - / dx e

u 0

(Of course, appropriate values for C and D are used, which satisfy the bounda-
ry conditions of symmetry around x ■ 0 and vanishing fields at x ■ *. ) One
may estimate from (25) by assuming, e.g., -Ej^q(x) " 100 mV/m and < ^ * 50
km which gives A^ •> 5 kV; this nominal value will decrease if Ej^q decreases
with X (as, for example, in (23)).

We see that A^ is determined in part by properties of magnetospheric
particles through ic (see the discussion below equation (2)), in part by iono-
spheric current conservation which couples neighboring field lines together,
and in part by perpendicular magnetospheric electric fields. If any of these
Ingredients is left out, it is not possible to determine the central potential
drop A^. Conversely, we can also say that A^ ^ 0 is the result of all these
Ingredients put together.

20

IV. SPECIFIC KKAMPCKS i)F ELECTRIC FIELD HAPPING

As we have indicated in the previous se< tli>n, our model suggests that the

magnitudes of the height-integrated Pedersen conductivity, E , and the paral-

P

lei current density, J|, directly affect the length scale associated with the
mapped electrostatic potential In the ionosphere. In this section we investi-
gate more closely the sensitivity of this mapping to latitudinal variations of
these parameters perpendicular to the magnetic field.

In the absence of conclusive observational descriptions of the spatial

variation of E , and J.; we have considered three kinds of variability which

p’ I

should bracket the physical characteristics we wish to model. Solutions to
(9) are discussed for three different assumptions; 1) F and < are spatially
constant; 2) F and tc assume constant values but < experiences a discontinuous
jump between regions of upgolng and downgoing current; 3) k is constant, but
F decreases exponi*tially as one goes from the precipitation region to the
return-current region (that is, E^ is enhanced in the precipitation region).
Of course, case 1) was discussed extensively in the last section.

Case 2) Mas also discussed briefly, in the special example of an exponen-
tial A somewhat more realistic is that used for Fig. 2, and given in

(22); it has no discontinuity in Ej^^(x) at x“0. Fig. 3 shows the results of

numerical integration of (14) for different with fixed at 75 km.

The most obvious result is a strong variation in the average return-current

potential drop, as necessitated by current conservation (the return current
2

varies as < 2 ^^ “ ® smaller requires a larger potential drop).

2 2

For < 2 “ 1<3k^ the return-current potential drop is 25 rimes smaller than the
central potential drop, and 5 times smaller for < 2 "* Note from Fig. 3 that

21

the central potential drop Increases slightly as 1 C 2 Increases, and that x,.
does not move very much*

So far, we have studiously avoided estimating the value of (that Is, Q

in the return~current region)^ This is because <2 easy to estimate

reliably, since it depends on quantities which vary significantly with alti-
tude in che ionosphere* But the reader is entitled to some feeling for the
ratio offer the following estimate. From Chiu and Cornwall

(1980], we recall

Ujjl - N^,_ e(2 + 3 ^)(e|A^J|/kTj^_)(kT,_/ 2 irm^)^^^ (2b)

From (b), we have

ijJ,| -Nj. .2

(27)

In (26) and (27), superscripts 1 and 2 refer to upward and downward currents,
respectively. The ratio of currents is thus

(2+A,/3) N„ m T,

TO

1/2

hv.

(2w)'

I- k T

1-

1^

(28)

Applying case W of Chiu and Schulz (1978) to (28), one has the magnetospheric
parameters: " 0.189 keV, hT^^_ ■ 0.775 keV, Nj^_ •• 3 cm”^. For iono-

o 2

spheric parameters, the lumped quantity hV|/Nj_ ~ 10^^ cm /sec for nighttime
conditions is used to obtain

uljl/lJ^jl » 0.2

(29)

22

If taken seriously, this Indicates that
E^/E^ is likely to be greater than one (because of precipitation enhance-
ment), It Is likely that - (Q^/Q^ )(£j /E^) U at least 10 and possibly

larger. This means (see Fig. 3) that return-current electrons have energies
of 100 eV or less.

Turn now to case 3), where the precipitation enhancement of E^ Is modeled
by an exponential, varying from 10 mho at x”0 to 1 mhn at 1 tree x. We have
fixed shown In Fig. 4, take the conductivity s. al<*

length to be either k ~^ or 0.5 ic ^ (also shown for comparison is the case E^ -
5 mho everywhere). The dynamo potential ia the same as for Figs. 2 and 3
(sea aquation (22)). The general features associated with the earlier figures
persist: The cross-over point x^, does not change much, and the return-current

potential drop Is significantly less than tl>a aantral potential drop. The
more rapid the falloff of E^, the larger this ratio of potential drops
bee ome s .

23

V.

ELECTROSTATIC POTENTIAL TOPOLOGIES: S AND V SHAPES

So far, we have only considered potentials ^ and ♦q which are symmetric
about x“0, with antisymmetric electric fields. This presumably is associated
with the classic inverted-V structure seen in auroral electron measurements
(Frank and Ackerson, 1971, 1972; Curnett, 1972; hizera et al., 197b; Mizera
and Fennell, 1977). But it has been suggested that asynunetric potential
structures (called S-shapes) may happen even more frequently than V-shapes.
V-shaped equipotentlal contours are always associated with Ej^ reversals, as
shown in Fig. 5. Whereas, the term "S-shape" refers to equipotentlal contours
which deviate from field lines, but do not show E^ reversals. Fig. 5 makes it
clear that every V-shaped (or symmetric) potential has S-shaped equipotentlal
contours on its wings. A satellite crossing this symmetric potential
structure at any altitude can detect a V-shaped region.

Bur this is not the only possibility in principle. There can be asym-
metric potential structures which look V-shaped at sufficiently high alti-
tudes, but are only S-shaped at lower altitudes. An example is shown in Fig.
b, in which the region of negative x has equipotentlal field lines, with an
auroral structure for x > 0. We do not know why the situation of Fig. b
should occur with any particular frequency, compared to the occurrence of more
or less symmetric potentials, but experimenters should keep in mind the possi-
bility that low-altitude electric field measurements might show a different
topology than seen on high-altitude satellites. It will be Interesting to see
what the Dynamics Explorer satellites see in this regard.

24

VI. CONCLUSIONS

1. We have extended Che differential equation used in an earlier work

[Chiu and Cornwall, 1980] to enrcmpass the return-current region. This re-
quires knowledge of the proportionality factor between -Jj and ^ which

we can estimate only crudely at the moment. (That Jj is proportional
to ♦ - is really r.othinp but Ohm's law, which is applicable to the return
current because it is generated in the colllsional ionosphere.) These esti-
mates suggest chat the return current, contiguous to and Just outside the
region of auroral precipitation, is carried by electrons of 100 eV or less.
The cross-over point between upward and downward current is 100-150 km from
the center of the inverted-V, for wide variety of auroral parameters. An
essential ingredient of this extended equation is the boundary condition of no
net current flow in or out of the ionosphere, so that the ionosphere is not
charged to grossly unphyslcal values. The ionosphere carries an extremely
small negative charge which is responsible for driving the return current.

2. Our differencial equation couples neighboring field lines to each
othtr. As a result, it is not possible to assign parallel potential drops
more or less arbitrarily, as earlier workers who did not consider ionospheric
current conservations were forced to do. The total parallel potential drop
along the center field line of an aurora Is uniquely determined by perpendicu-

magnetospherlc electric fields, convolved with a Green's function which
has a scale length x ^ determined by both ionospheric parameters and by the
number and momentum of auroral primaries: x ■ Q/lp, Q " Ne^/Mv. This

unique determination of parallel potentials means that the problem of con-
structing ^ (both and E^) everywhere in the magnetosphere, given E^ on a
boundary surface can be solved straightforwardly (In principle, at least).

25

3. The Green'e function integral* which solve our differential equation

show that small-scale structure* In Ej^q<*). the equatorial magnetospherlc
field, ate mapped onto the Ionosphere. But large-scale structure is hidden,
and the overall Ionospheric scale slse of Inverted— V auroras Is k ■ 100
kra. »> may be called the outer scale slae of Inverted V's. There Is, of
course, much small-scale structure In auroral arcs, and many authors b«Heve
that It ts Trapped down from small-scale structures created near the equator of
auroral field lines.

4. Different boundary conditions Imposed on the differential equation
yield topologlcally-distlnct solutions. Latitudinal symmetry about a center
line implies equipotential contours with a V shape at all altitudes
(sufficiently close to the center line). Asymmetric boundary conditions can
push the V-shaped region to a finite altitude, leaving only S-shaped
potentials below. The Dynamics Explorer satellites will, we hope, settle this
question of the nature of V-shapes and S-shapes.

^6

APPENDIX

In this Appendix %>e show that the kinetic Ohts's lew (2) holds for
arbitrary electron distribution function.

2 2

Let - VjQ and s be the constants of notion for an electron
moving on a magnetic field line; these are related to the local velocities at
any point s on the line by

^o“ ''i ^ ■ T " ♦o^

''io " ITiT ''i

The equatorial distribution function (s*0, B~Bq, ^~4q) la f(C|> Cj^)*

We are interested in the current at the ionosphere (s*t) produced by
electrons arriving there from the equator. The velocity-space integral which
defines J| is subject to the constraints

V, > 0, y,] > 0 (A-3)

which translates to the constraints

- Jl

— ■ 2’' Jo"''! ‘*'’1 f “

e

> e(c,-R) etc, - (-i; — {c,-R)l}

I Bq 1

(A-4)

27

where 6 is the usual step function and

” M - bj ^^4 " *0^ ^ °

4 0

Now Bq/B^ < 1 so that If, as we assume, the average value of is not large
compared to (e/M) - ^q), we can replace by 0 in f for the

terms 0(R-ej^) in (A-A). Likewise in the second 9-function in (A-4) we can
set R“0, so 0 ^ ^ then straightforward to find

■ I /o’«l ^

e

The E| term in square brackets represents the diffuse auroral current (leakage

into the loss cone), and will be neglected in this work. The second terra

2 —

gives rise to equation (2), where comparison with (A-6) shows that Q • N /Mv

e

where v is a typical velocity for f.

28

Acknowledgment

The authors acknowledge beneficial discussions on various aspects of this
paper with S. ~I. Akasofu, D. S. Evans, J. F. Fennell, D. J. Gorney, M. K.
Hudson, J. R. Kan, J. Lemalre, L. R. Lyons, R* L. Lysak, P* F* Mlzera, M.
Schulz and R. R. Vondrak.

29

REFERENCES

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3Z

Figure Captions

1. Electrostatic field mapping along field lines and L 2 between the

magnetos pherlc equator &nd the Ionosphere (E^) with or without a

magnetic field-aligned potential drop

2. Latitudinal structure of the electrostatic potential associated with the
magnetospherlc boundary, and with the Ionospheric boundary, x,. is
the nosltlon at which ^

3. Latitudinal structure of ^ and from (9) with k •• for x < x^ and <

■ <2 “ ^ *c* Variations In m allow different values of in

regions of upgolng and downgoing current (F ■ 1).

A. Latitudinal structure of ^ and for an exponentially varying Integrated
Pedersen conductivity, 9.1e + 0,9, ( < ■ Is kept constant,)

5, Interpolated equlpotentlal structure from the geometrically mapped mag-
netospherlc boundary to the Ionosphere. Arrows Indicate the direction of
the electric field. Notations V and S Indicate regions of V-shaped and
S-shaped potential structure,

6, Same as Fig. 3 for a different imposed magnetospherlc potential struc-
ture.

33

POLAR DISI